2018 Fiscal Year Annual Research Report
Analysis of counting and enumeration problems pertaining to physically realistic biological systems
Project/Area Number 
18F18117

Research Institution  The University of Tokyo 
Host Researcher 
陶山 明 東京大学, 大学院総合文化研究科, 教授 (90163063)

Foreign Research Fellow 
BARISH ROBERT 東京大学, 総合文化研究科, 外国人特別研究員

Project Period (FY) 
20180425 – 20200331

Keywords  Counting complexity / Graph theory / Combinatorics / Network analysis / Cycles / Paths / 2Factors 
Outline of Annual Research Achievements 
Towards a characterization of the complexity of cycle and path counting problems on graph classes arising in the context of, for example, biological network analysis and mathematical physics, we have shown (under standard complexity theoretic assumptions) that no combination of planarity, bipartiteness, vertex degree, and vertex connectivity constraints can conspire to allow for exact or approximate counting of Hamiltonian cycles, Hamiltonian paths, simple cycles, or simple paths in polynomial time. We also report significant progress on a program of using Courcelle's wellknown metatheorem to automatically extend these intractability results to a large number of other natural graph classes. In the process of these efforts we have made a number of more general mathematical accomplishments, for example, in disproving a circa 2011 conjecture of Gordon et. al. that the Hamiltonian cycle decision problem on locally connected graphs of vertex degree at most 6 is efficiently solvable.
Concerning tractability results, for graph classes of bounded pathwidth we have established general methods for finding and proving generating functions, and where possible, analytic expressions, for counting Hamiltonian cycles, Hamiltonian paths, simple cycles, and simple paths. These methods succeed even in highcomplexity cases, such as for infinite families of sliced recursive triangulations which arise in the context of mathematical physics models (e.g. in Lorentzian models). We have presented these findings at international mathematical physics conferences.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
As expected, we were able to characterize the hardness of counting and approximately counting Hamiltonian cycles and Hamiltonian paths on upwards of approximately 800 graph classes, including all 733 cases where there exists an NPcompleteness result for the Hamiltonian cycle decision problem. For the majority of these cases, we have likewise been able to characterize the hardness of counting and approximately counting simple cycles and simple paths. Much more unexpectedly, however, was the fact that we were able to use Courcelle's wellknown metatheorem to automate the process of establishing these hardness results for approximately 424 of aforementioned graph classes.
Another unexpected development was our ability to largely automate the process of finding and proving generating functions, and where possible, analytic expressions, for exactly counting Hamiltonian cycles, Hamiltonian paths, simple cycles, and simple paths on bounded pathwidth graphs of sufficiently high complexity to be of relevance to the mathematical physics community.
Additionally, we found a number of surprising connections between our work on cycle and path counting and the closely related problem of counting what are known as 2factors of a graph (i.e. 2regular covering subgraphs). These are related to what are known as fullypacked loop configurations in the mathematical physics community, and have close connections to sixvertex or icetype models in statistical mechanics. We have already obtained a number of results for 2factor counting problems on graph classes of interest to physicists.

Strategy for Future Research Activity 
We will finalize and publish a number of manuscripts in preparation from 2018. This includes the proof, discussed in our summary of research achievements for FY2018, that no combination of planarity, bipartiteness, vertex degree, and vertex connectivity constraints can conspire to allow for exact or approximate counting of Hamiltonian cycles, Hamiltonian paths, simple cycles, or simple paths in polynomial time.
We will work towards completing our characterization of the complexity of exactly and approximately counting Hamiltonian cycles, Hamiltonian paths, simple cycles, and simple paths on all approximately 1246 classes of graphs in the ISGCI, or to show that these results would imply the solution to wellknown open problems in the literature. On the basis of our accomplishments this year, we feel that this objective is within reach.
Additionally, as stated in the current status of the research section, there exist significant connections between our work on counting cycles and paths and the problem in mathematical physics of counting 2factors on various types of graphs. Unfortunately, however, almost all complexity theoretic results for counting 2factors are via reduction from the problem of counting Eulerian circuits, for which very little appears to be known. we will make an effort to build a bridge between our cycle and path counting results and 2factor counting problems on a wide variety of graph classes, which if possible, would substantially improve the situation.

Research Products
(3 results)